The azimuthal anisotropy Fourier coefficients (v_{n}) in 8.16 TeV p+Pb data are extracted via long-range two-particle correlations as a function of the event multiplicity and compared to corresponding results in pp and PbPb collisions. Using a four-particle cumulant technique, v_{n} correlations are measured for the first time in pp and p+Pb collisions. The v_{2} and v_{4} coefficients are found to be positively correlated in all collision systems. For high-multiplicity p+Pb collisions, an anticorrelation of v_{2} and v_{3} is observed, with a similar correlation strength as in PbPb data at the same multiplicity. The new correlation results strengthen the case for a common origin of the collectivity seen in p+Pb and PbPb collisions in the measured multiplicity range.